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Full text of "Treatise On Analysis Vol-Ii"

432        XV   NORMED ALGEBRAS AND SPECTRAL THEORY

(15.13.7)    The defect of a closed Hermitian operator H is defined to be the
pair (m, ri) where m (resp. n) is equal to the dimension of E^ (resp. E#) if this
dimension is finite, and is +00 otherwise; m is called the positive defect and n
the negative defect of H.

(15.13.8)    For a closed hermitian operator H to be self-adjoint, it is necessary
and sufficient that its defect should be (0, 0), or in other words that its Cayley
transform V should be unitary. For H to be capable of extension to a self-
adjoint operator, it is necessary and sufficient that its defect should be of the
form (m, m).

The first assertion follows trivially from (15.13.6). Next, if H is the re-
striction of a self-adjoint operator A, then the Cayley transform V of H is the
restriction of the Cayley transform UofA, which is unitary. Hence by defini-
tion we have E# = J7(E#) because U is an automorphism of the Hilbert
space E, and the defect of H is therefore of the form (m, m). Conversely,
if this is so, then there exists an automorphism U of the Hilbert space E
which coincides with V on dom(F) and is such that U(E) = E# (because
two Hilbert spaces of the same finite dimension, or two separable infinite-
dimensional Hilbert spaces, are isomorphic (6.6.2)). Since U is a unitary
operator on E, and the image of E under I U (which contains dom(HJ) is
dense in E, it follows that (7 is the Cayley transform of a self-adjoint operator
which extends H.

A hermitian operator H whose closure H** is self-adjoint is said to be
essentially self-adjoint.

Examples

(15.13.9) Let (On^o be a Hilbert basis of an infinite-dimensional separ-
able Hilbert space E. Let  be the closed hyperplane in E generated by the
en with n *z 1 (the orthogonal supplement of the line Ce0) and let U' be the
isometry of F onto E such that U'  en = en.1 for all n ;> 1. Then the image G'
of / U' is dense in E. For G' contains the vectors en  en+1 for all n ^ 0,

oo

and if a vector x=  en in E is orthogonal to all the vectors em - em+ (m *> 0),

w = 0

then we have m = m+1 for all m ^ 0, so that all the  (n ^ 0) are equal; and

00

this is not compatible with the convergence of the series  |J2 unless

=o) onto r(domCO), such that r= VR. (As in the proof of (15.12.8),